Properties

Label 6762.2.a.cj.1.4
Level $6762$
Weight $2$
Character 6762.1
Self dual yes
Analytic conductor $53.995$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 6762 = 2 \cdot 3 \cdot 7^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6762.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(53.9948418468\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.23252.1
Defining polynomial: \(x^{4} - x^{3} - 6 x^{2} + 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(-0.634868\) of defining polynomial
Character \(\chi\) \(=\) 6762.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +3.47747 q^{5} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +3.47747 q^{5} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +3.47747 q^{10} -1.26974 q^{11} -1.00000 q^{12} -5.47747 q^{13} -3.47747 q^{15} +1.00000 q^{16} -7.03079 q^{17} +1.00000 q^{18} -2.00000 q^{19} +3.47747 q^{20} -1.26974 q^{22} -1.00000 q^{23} -1.00000 q^{24} +7.09279 q^{25} -5.47747 q^{26} -1.00000 q^{27} -7.40162 q^{29} -3.47747 q^{30} +9.57026 q^{31} +1.00000 q^{32} +1.26974 q^{33} -7.03079 q^{34} +1.00000 q^{36} +1.71642 q^{37} -2.00000 q^{38} +5.47747 q^{39} +3.47747 q^{40} +8.50826 q^{41} -7.23852 q^{43} -1.26974 q^{44} +3.47747 q^{45} -1.00000 q^{46} -12.0169 q^{47} -1.00000 q^{48} +7.09279 q^{50} +7.03079 q^{51} -5.47747 q^{52} +7.68520 q^{53} -1.00000 q^{54} -4.41547 q^{55} +2.00000 q^{57} -7.40162 q^{58} -7.84830 q^{59} -3.47747 q^{60} -9.92415 q^{61} +9.57026 q^{62} +1.00000 q^{64} -19.0477 q^{65} +1.26974 q^{66} -4.37637 q^{67} -7.03079 q^{68} +1.00000 q^{69} -10.0616 q^{71} +1.00000 q^{72} -13.4944 q^{73} +1.71642 q^{74} -7.09279 q^{75} -2.00000 q^{76} +5.47747 q^{78} +13.4944 q^{79} +3.47747 q^{80} +1.00000 q^{81} +8.50826 q^{82} -5.10664 q^{83} -24.4493 q^{85} -7.23852 q^{86} +7.40162 q^{87} -1.26974 q^{88} -11.0308 q^{89} +3.47747 q^{90} -1.00000 q^{92} -9.57026 q^{93} -12.0169 q^{94} -6.95494 q^{95} -1.00000 q^{96} -14.5564 q^{97} -1.26974 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 4q^{2} - 4q^{3} + 4q^{4} - 3q^{5} - 4q^{6} + 4q^{8} + 4q^{9} + O(q^{10}) \) \( 4q + 4q^{2} - 4q^{3} + 4q^{4} - 3q^{5} - 4q^{6} + 4q^{8} + 4q^{9} - 3q^{10} + 2q^{11} - 4q^{12} - 5q^{13} + 3q^{15} + 4q^{16} - 10q^{17} + 4q^{18} - 8q^{19} - 3q^{20} + 2q^{22} - 4q^{23} - 4q^{24} + 13q^{25} - 5q^{26} - 4q^{27} + 3q^{29} + 3q^{30} + 6q^{31} + 4q^{32} - 2q^{33} - 10q^{34} + 4q^{36} + q^{37} - 8q^{38} + 5q^{39} - 3q^{40} - q^{41} - q^{43} + 2q^{44} - 3q^{45} - 4q^{46} - 17q^{47} - 4q^{48} + 13q^{50} + 10q^{51} - 5q^{52} + 4q^{53} - 4q^{54} + 2q^{55} + 8q^{57} + 3q^{58} + 3q^{60} - 24q^{61} + 6q^{62} + 4q^{64} - 27q^{65} - 2q^{66} - 8q^{67} - 10q^{68} + 4q^{69} - 4q^{71} + 4q^{72} - 6q^{73} + q^{74} - 13q^{75} - 8q^{76} + 5q^{78} + 6q^{79} - 3q^{80} + 4q^{81} - q^{82} - 18q^{83} - 16q^{85} - q^{86} - 3q^{87} + 2q^{88} - 26q^{89} - 3q^{90} - 4q^{92} - 6q^{93} - 17q^{94} + 6q^{95} - 4q^{96} - 13q^{97} + 2q^{99} + O(q^{100}) \)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) −1.00000 −0.577350
\(4\) 1.00000 0.500000
\(5\) 3.47747 1.55517 0.777586 0.628777i \(-0.216444\pi\)
0.777586 + 0.628777i \(0.216444\pi\)
\(6\) −1.00000 −0.408248
\(7\) 0 0
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 3.47747 1.09967
\(11\) −1.26974 −0.382840 −0.191420 0.981508i \(-0.561309\pi\)
−0.191420 + 0.981508i \(0.561309\pi\)
\(12\) −1.00000 −0.288675
\(13\) −5.47747 −1.51918 −0.759588 0.650404i \(-0.774600\pi\)
−0.759588 + 0.650404i \(0.774600\pi\)
\(14\) 0 0
\(15\) −3.47747 −0.897879
\(16\) 1.00000 0.250000
\(17\) −7.03079 −1.70522 −0.852608 0.522551i \(-0.824981\pi\)
−0.852608 + 0.522551i \(0.824981\pi\)
\(18\) 1.00000 0.235702
\(19\) −2.00000 −0.458831 −0.229416 0.973329i \(-0.573682\pi\)
−0.229416 + 0.973329i \(0.573682\pi\)
\(20\) 3.47747 0.777586
\(21\) 0 0
\(22\) −1.26974 −0.270709
\(23\) −1.00000 −0.208514
\(24\) −1.00000 −0.204124
\(25\) 7.09279 1.41856
\(26\) −5.47747 −1.07422
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −7.40162 −1.37445 −0.687223 0.726446i \(-0.741171\pi\)
−0.687223 + 0.726446i \(0.741171\pi\)
\(30\) −3.47747 −0.634896
\(31\) 9.57026 1.71887 0.859435 0.511246i \(-0.170816\pi\)
0.859435 + 0.511246i \(0.170816\pi\)
\(32\) 1.00000 0.176777
\(33\) 1.26974 0.221033
\(34\) −7.03079 −1.20577
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 1.71642 0.282177 0.141089 0.989997i \(-0.454940\pi\)
0.141089 + 0.989997i \(0.454940\pi\)
\(38\) −2.00000 −0.324443
\(39\) 5.47747 0.877097
\(40\) 3.47747 0.549836
\(41\) 8.50826 1.32877 0.664383 0.747392i \(-0.268694\pi\)
0.664383 + 0.747392i \(0.268694\pi\)
\(42\) 0 0
\(43\) −7.23852 −1.10386 −0.551932 0.833889i \(-0.686109\pi\)
−0.551932 + 0.833889i \(0.686109\pi\)
\(44\) −1.26974 −0.191420
\(45\) 3.47747 0.518390
\(46\) −1.00000 −0.147442
\(47\) −12.0169 −1.75285 −0.876425 0.481538i \(-0.840078\pi\)
−0.876425 + 0.481538i \(0.840078\pi\)
\(48\) −1.00000 −0.144338
\(49\) 0 0
\(50\) 7.09279 1.00307
\(51\) 7.03079 0.984507
\(52\) −5.47747 −0.759588
\(53\) 7.68520 1.05564 0.527822 0.849355i \(-0.323009\pi\)
0.527822 + 0.849355i \(0.323009\pi\)
\(54\) −1.00000 −0.136083
\(55\) −4.41547 −0.595381
\(56\) 0 0
\(57\) 2.00000 0.264906
\(58\) −7.40162 −0.971880
\(59\) −7.84830 −1.02176 −0.510881 0.859652i \(-0.670681\pi\)
−0.510881 + 0.859652i \(0.670681\pi\)
\(60\) −3.47747 −0.448939
\(61\) −9.92415 −1.27066 −0.635329 0.772242i \(-0.719135\pi\)
−0.635329 + 0.772242i \(0.719135\pi\)
\(62\) 9.57026 1.21542
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −19.0477 −2.36258
\(66\) 1.26974 0.156294
\(67\) −4.37637 −0.534659 −0.267330 0.963605i \(-0.586141\pi\)
−0.267330 + 0.963605i \(0.586141\pi\)
\(68\) −7.03079 −0.852608
\(69\) 1.00000 0.120386
\(70\) 0 0
\(71\) −10.0616 −1.19409 −0.597045 0.802208i \(-0.703659\pi\)
−0.597045 + 0.802208i \(0.703659\pi\)
\(72\) 1.00000 0.117851
\(73\) −13.4944 −1.57940 −0.789701 0.613492i \(-0.789764\pi\)
−0.789701 + 0.613492i \(0.789764\pi\)
\(74\) 1.71642 0.199529
\(75\) −7.09279 −0.819005
\(76\) −2.00000 −0.229416
\(77\) 0 0
\(78\) 5.47747 0.620201
\(79\) 13.4944 1.51824 0.759120 0.650951i \(-0.225630\pi\)
0.759120 + 0.650951i \(0.225630\pi\)
\(80\) 3.47747 0.388793
\(81\) 1.00000 0.111111
\(82\) 8.50826 0.939580
\(83\) −5.10664 −0.560526 −0.280263 0.959923i \(-0.590422\pi\)
−0.280263 + 0.959923i \(0.590422\pi\)
\(84\) 0 0
\(85\) −24.4493 −2.65190
\(86\) −7.23852 −0.780550
\(87\) 7.40162 0.793537
\(88\) −1.26974 −0.135354
\(89\) −11.0308 −1.16926 −0.584631 0.811300i \(-0.698761\pi\)
−0.584631 + 0.811300i \(0.698761\pi\)
\(90\) 3.47747 0.366557
\(91\) 0 0
\(92\) −1.00000 −0.104257
\(93\) −9.57026 −0.992390
\(94\) −12.0169 −1.23945
\(95\) −6.95494 −0.713562
\(96\) −1.00000 −0.102062
\(97\) −14.5564 −1.47798 −0.738990 0.673717i \(-0.764697\pi\)
−0.738990 + 0.673717i \(0.764697\pi\)
\(98\) 0 0
\(99\) −1.26974 −0.127613
\(100\) 7.09279 0.709279
\(101\) 0.817512 0.0813455 0.0406727 0.999173i \(-0.487050\pi\)
0.0406727 + 0.999173i \(0.487050\pi\)
\(102\) 7.03079 0.696152
\(103\) 4.50826 0.444212 0.222106 0.975023i \(-0.428707\pi\)
0.222106 + 0.975023i \(0.428707\pi\)
\(104\) −5.47747 −0.537110
\(105\) 0 0
\(106\) 7.68520 0.746453
\(107\) −3.62363 −0.350309 −0.175155 0.984541i \(-0.556043\pi\)
−0.175155 + 0.984541i \(0.556043\pi\)
\(108\) −1.00000 −0.0962250
\(109\) 18.3175 1.75449 0.877247 0.480038i \(-0.159377\pi\)
0.877247 + 0.480038i \(0.159377\pi\)
\(110\) −4.41547 −0.420998
\(111\) −1.71642 −0.162915
\(112\) 0 0
\(113\) −10.5083 −0.988534 −0.494267 0.869310i \(-0.664563\pi\)
−0.494267 + 0.869310i \(0.664563\pi\)
\(114\) 2.00000 0.187317
\(115\) −3.47747 −0.324276
\(116\) −7.40162 −0.687223
\(117\) −5.47747 −0.506392
\(118\) −7.84830 −0.722495
\(119\) 0 0
\(120\) −3.47747 −0.317448
\(121\) −9.38777 −0.853434
\(122\) −9.92415 −0.898490
\(123\) −8.50826 −0.767164
\(124\) 9.57026 0.859435
\(125\) 7.27761 0.650930
\(126\) 0 0
\(127\) 11.6149 1.03066 0.515328 0.856993i \(-0.327670\pi\)
0.515328 + 0.856993i \(0.327670\pi\)
\(128\) 1.00000 0.0883883
\(129\) 7.23852 0.637316
\(130\) −19.0477 −1.67060
\(131\) 19.1405 1.67231 0.836157 0.548489i \(-0.184797\pi\)
0.836157 + 0.548489i \(0.184797\pi\)
\(132\) 1.26974 0.110516
\(133\) 0 0
\(134\) −4.37637 −0.378061
\(135\) −3.47747 −0.299293
\(136\) −7.03079 −0.602885
\(137\) 4.86215 0.415401 0.207701 0.978192i \(-0.433402\pi\)
0.207701 + 0.978192i \(0.433402\pi\)
\(138\) 1.00000 0.0851257
\(139\) 16.5083 1.40021 0.700106 0.714039i \(-0.253136\pi\)
0.700106 + 0.714039i \(0.253136\pi\)
\(140\) 0 0
\(141\) 12.0169 1.01201
\(142\) −10.0616 −0.844349
\(143\) 6.95494 0.581601
\(144\) 1.00000 0.0833333
\(145\) −25.7389 −2.13750
\(146\) −13.4944 −1.11681
\(147\) 0 0
\(148\) 1.71642 0.141089
\(149\) 18.8257 1.54226 0.771132 0.636676i \(-0.219691\pi\)
0.771132 + 0.636676i \(0.219691\pi\)
\(150\) −7.09279 −0.579124
\(151\) 15.4632 1.25838 0.629188 0.777253i \(-0.283387\pi\)
0.629188 + 0.777253i \(0.283387\pi\)
\(152\) −2.00000 −0.162221
\(153\) −7.03079 −0.568406
\(154\) 0 0
\(155\) 33.2803 2.67314
\(156\) 5.47747 0.438548
\(157\) −12.8791 −1.02786 −0.513932 0.857831i \(-0.671811\pi\)
−0.513932 + 0.857831i \(0.671811\pi\)
\(158\) 13.4944 1.07356
\(159\) −7.68520 −0.609476
\(160\) 3.47747 0.274918
\(161\) 0 0
\(162\) 1.00000 0.0785674
\(163\) −23.1405 −1.81251 −0.906253 0.422736i \(-0.861070\pi\)
−0.906253 + 0.422736i \(0.861070\pi\)
\(164\) 8.50826 0.664383
\(165\) 4.41547 0.343744
\(166\) −5.10664 −0.396352
\(167\) 17.4186 1.34789 0.673944 0.738782i \(-0.264599\pi\)
0.673944 + 0.738782i \(0.264599\pi\)
\(168\) 0 0
\(169\) 17.0027 1.30790
\(170\) −24.4493 −1.87518
\(171\) −2.00000 −0.152944
\(172\) −7.23852 −0.551932
\(173\) −2.84521 −0.216317 −0.108159 0.994134i \(-0.534495\pi\)
−0.108159 + 0.994134i \(0.534495\pi\)
\(174\) 7.40162 0.561115
\(175\) 0 0
\(176\) −1.26974 −0.0957099
\(177\) 7.84830 0.589914
\(178\) −11.0308 −0.826793
\(179\) 1.13785 0.0850471 0.0425235 0.999095i \(-0.486460\pi\)
0.0425235 + 0.999095i \(0.486460\pi\)
\(180\) 3.47747 0.259195
\(181\) −18.9407 −1.40785 −0.703924 0.710275i \(-0.748571\pi\)
−0.703924 + 0.710275i \(0.748571\pi\)
\(182\) 0 0
\(183\) 9.92415 0.733614
\(184\) −1.00000 −0.0737210
\(185\) 5.96879 0.438834
\(186\) −9.57026 −0.701725
\(187\) 8.92724 0.652825
\(188\) −12.0169 −0.876425
\(189\) 0 0
\(190\) −6.95494 −0.504564
\(191\) −0.151700 −0.0109766 −0.00548832 0.999985i \(-0.501747\pi\)
−0.00548832 + 0.999985i \(0.501747\pi\)
\(192\) −1.00000 −0.0721688
\(193\) 10.9237 0.786307 0.393153 0.919473i \(-0.371384\pi\)
0.393153 + 0.919473i \(0.371384\pi\)
\(194\) −14.5564 −1.04509
\(195\) 19.0477 1.36404
\(196\) 0 0
\(197\) 13.6149 0.970021 0.485011 0.874508i \(-0.338816\pi\)
0.485011 + 0.874508i \(0.338816\pi\)
\(198\) −1.26974 −0.0902362
\(199\) −22.5698 −1.59993 −0.799967 0.600045i \(-0.795149\pi\)
−0.799967 + 0.600045i \(0.795149\pi\)
\(200\) 7.09279 0.501536
\(201\) 4.37637 0.308686
\(202\) 0.817512 0.0575199
\(203\) 0 0
\(204\) 7.03079 0.492254
\(205\) 29.5872 2.06646
\(206\) 4.50826 0.314105
\(207\) −1.00000 −0.0695048
\(208\) −5.47747 −0.379794
\(209\) 2.53947 0.175659
\(210\) 0 0
\(211\) 9.49441 0.653622 0.326811 0.945090i \(-0.394026\pi\)
0.326811 + 0.945090i \(0.394026\pi\)
\(212\) 7.68520 0.527822
\(213\) 10.0616 0.689408
\(214\) −3.62363 −0.247706
\(215\) −25.1717 −1.71670
\(216\) −1.00000 −0.0680414
\(217\) 0 0
\(218\) 18.3175 1.24062
\(219\) 13.4944 0.911868
\(220\) −4.41547 −0.297691
\(221\) 38.5109 2.59053
\(222\) −1.71642 −0.115198
\(223\) −15.2946 −1.02420 −0.512100 0.858926i \(-0.671132\pi\)
−0.512100 + 0.858926i \(0.671132\pi\)
\(224\) 0 0
\(225\) 7.09279 0.472853
\(226\) −10.5083 −0.698999
\(227\) −1.42931 −0.0948669 −0.0474335 0.998874i \(-0.515104\pi\)
−0.0474335 + 0.998874i \(0.515104\pi\)
\(228\) 2.00000 0.132453
\(229\) −17.9241 −1.18446 −0.592231 0.805769i \(-0.701753\pi\)
−0.592231 + 0.805769i \(0.701753\pi\)
\(230\) −3.47747 −0.229298
\(231\) 0 0
\(232\) −7.40162 −0.485940
\(233\) −18.4155 −1.20644 −0.603219 0.797576i \(-0.706115\pi\)
−0.603219 + 0.797576i \(0.706115\pi\)
\(234\) −5.47747 −0.358073
\(235\) −41.7885 −2.72598
\(236\) −7.84830 −0.510881
\(237\) −13.4944 −0.876556
\(238\) 0 0
\(239\) −8.00000 −0.517477 −0.258738 0.965947i \(-0.583307\pi\)
−0.258738 + 0.965947i \(0.583307\pi\)
\(240\) −3.47747 −0.224470
\(241\) 4.39853 0.283334 0.141667 0.989914i \(-0.454754\pi\)
0.141667 + 0.989914i \(0.454754\pi\)
\(242\) −9.38777 −0.603469
\(243\) −1.00000 −0.0641500
\(244\) −9.92415 −0.635329
\(245\) 0 0
\(246\) −8.50826 −0.542467
\(247\) 10.9549 0.697046
\(248\) 9.57026 0.607712
\(249\) 5.10664 0.323620
\(250\) 7.27761 0.460277
\(251\) −4.29498 −0.271097 −0.135548 0.990771i \(-0.543280\pi\)
−0.135548 + 0.990771i \(0.543280\pi\)
\(252\) 0 0
\(253\) 1.26974 0.0798276
\(254\) 11.6149 0.728783
\(255\) 24.4493 1.53108
\(256\) 1.00000 0.0625000
\(257\) −13.0451 −0.813729 −0.406864 0.913489i \(-0.633378\pi\)
−0.406864 + 0.913489i \(0.633378\pi\)
\(258\) 7.23852 0.450651
\(259\) 0 0
\(260\) −19.0477 −1.18129
\(261\) −7.40162 −0.458149
\(262\) 19.1405 1.18251
\(263\) 13.4016 0.826379 0.413190 0.910645i \(-0.364415\pi\)
0.413190 + 0.910645i \(0.364415\pi\)
\(264\) 1.26974 0.0781468
\(265\) 26.7251 1.64171
\(266\) 0 0
\(267\) 11.0308 0.675073
\(268\) −4.37637 −0.267330
\(269\) 9.09236 0.554371 0.277186 0.960816i \(-0.410598\pi\)
0.277186 + 0.960816i \(0.410598\pi\)
\(270\) −3.47747 −0.211632
\(271\) −15.9241 −0.967323 −0.483662 0.875255i \(-0.660693\pi\)
−0.483662 + 0.875255i \(0.660693\pi\)
\(272\) −7.03079 −0.426304
\(273\) 0 0
\(274\) 4.86215 0.293733
\(275\) −9.00597 −0.543080
\(276\) 1.00000 0.0601929
\(277\) −18.7527 −1.12674 −0.563372 0.826204i \(-0.690496\pi\)
−0.563372 + 0.826204i \(0.690496\pi\)
\(278\) 16.5083 0.990099
\(279\) 9.57026 0.572956
\(280\) 0 0
\(281\) −24.4806 −1.46039 −0.730194 0.683240i \(-0.760570\pi\)
−0.730194 + 0.683240i \(0.760570\pi\)
\(282\) 12.0169 0.715598
\(283\) 6.18558 0.367695 0.183847 0.982955i \(-0.441145\pi\)
0.183847 + 0.982955i \(0.441145\pi\)
\(284\) −10.0616 −0.597045
\(285\) 6.95494 0.411975
\(286\) 6.95494 0.411254
\(287\) 0 0
\(288\) 1.00000 0.0589256
\(289\) 32.4320 1.90776
\(290\) −25.7389 −1.51144
\(291\) 14.5564 0.853312
\(292\) −13.4944 −0.789701
\(293\) 32.0196 1.87061 0.935303 0.353849i \(-0.115127\pi\)
0.935303 + 0.353849i \(0.115127\pi\)
\(294\) 0 0
\(295\) −27.2922 −1.58901
\(296\) 1.71642 0.0997647
\(297\) 1.26974 0.0736775
\(298\) 18.8257 1.09054
\(299\) 5.47747 0.316770
\(300\) −7.09279 −0.409502
\(301\) 0 0
\(302\) 15.4632 0.889807
\(303\) −0.817512 −0.0469648
\(304\) −2.00000 −0.114708
\(305\) −34.5109 −1.97609
\(306\) −7.03079 −0.401923
\(307\) 20.1710 1.15122 0.575609 0.817725i \(-0.304765\pi\)
0.575609 + 0.817725i \(0.304765\pi\)
\(308\) 0 0
\(309\) −4.50826 −0.256466
\(310\) 33.2803 1.89019
\(311\) −8.10973 −0.459861 −0.229930 0.973207i \(-0.573850\pi\)
−0.229930 + 0.973207i \(0.573850\pi\)
\(312\) 5.47747 0.310101
\(313\) −6.84521 −0.386914 −0.193457 0.981109i \(-0.561970\pi\)
−0.193457 + 0.981109i \(0.561970\pi\)
\(314\) −12.8791 −0.726809
\(315\) 0 0
\(316\) 13.4944 0.759120
\(317\) −0.598381 −0.0336084 −0.0168042 0.999859i \(-0.505349\pi\)
−0.0168042 + 0.999859i \(0.505349\pi\)
\(318\) −7.68520 −0.430965
\(319\) 9.39810 0.526193
\(320\) 3.47747 0.194396
\(321\) 3.62363 0.202251
\(322\) 0 0
\(323\) 14.0616 0.782407
\(324\) 1.00000 0.0555556
\(325\) −38.8505 −2.15504
\(326\) −23.1405 −1.28163
\(327\) −18.3175 −1.01296
\(328\) 8.50826 0.469790
\(329\) 0 0
\(330\) 4.41547 0.243063
\(331\) 25.9099 1.42414 0.712068 0.702111i \(-0.247759\pi\)
0.712068 + 0.702111i \(0.247759\pi\)
\(332\) −5.10664 −0.280263
\(333\) 1.71642 0.0940591
\(334\) 17.4186 0.953101
\(335\) −15.2187 −0.831487
\(336\) 0 0
\(337\) −8.21328 −0.447406 −0.223703 0.974657i \(-0.571815\pi\)
−0.223703 + 0.974657i \(0.571815\pi\)
\(338\) 17.0027 0.924823
\(339\) 10.5083 0.570730
\(340\) −24.4493 −1.32595
\(341\) −12.1517 −0.658051
\(342\) −2.00000 −0.108148
\(343\) 0 0
\(344\) −7.23852 −0.390275
\(345\) 3.47747 0.187221
\(346\) −2.84521 −0.152959
\(347\) −15.4632 −0.830108 −0.415054 0.909797i \(-0.636237\pi\)
−0.415054 + 0.909797i \(0.636237\pi\)
\(348\) 7.40162 0.396768
\(349\) 11.9241 0.638285 0.319143 0.947707i \(-0.396605\pi\)
0.319143 + 0.947707i \(0.396605\pi\)
\(350\) 0 0
\(351\) 5.47747 0.292366
\(352\) −1.26974 −0.0676771
\(353\) −13.9688 −0.743483 −0.371742 0.928336i \(-0.621239\pi\)
−0.371742 + 0.928336i \(0.621239\pi\)
\(354\) 7.84830 0.417132
\(355\) −34.9888 −1.85701
\(356\) −11.0308 −0.584631
\(357\) 0 0
\(358\) 1.13785 0.0601374
\(359\) 6.56983 0.346743 0.173371 0.984857i \(-0.444534\pi\)
0.173371 + 0.984857i \(0.444534\pi\)
\(360\) 3.47747 0.183279
\(361\) −15.0000 −0.789474
\(362\) −18.9407 −0.995499
\(363\) 9.38777 0.492730
\(364\) 0 0
\(365\) −46.9264 −2.45624
\(366\) 9.92415 0.518744
\(367\) 13.2499 0.691640 0.345820 0.938301i \(-0.387601\pi\)
0.345820 + 0.938301i \(0.387601\pi\)
\(368\) −1.00000 −0.0521286
\(369\) 8.50826 0.442922
\(370\) 5.96879 0.310302
\(371\) 0 0
\(372\) −9.57026 −0.496195
\(373\) 19.6852 1.01926 0.509631 0.860393i \(-0.329782\pi\)
0.509631 + 0.860393i \(0.329782\pi\)
\(374\) 8.92724 0.461617
\(375\) −7.27761 −0.375814
\(376\) −12.0169 −0.619726
\(377\) 40.5421 2.08803
\(378\) 0 0
\(379\) 9.77799 0.502262 0.251131 0.967953i \(-0.419198\pi\)
0.251131 + 0.967953i \(0.419198\pi\)
\(380\) −6.95494 −0.356781
\(381\) −11.6149 −0.595049
\(382\) −0.151700 −0.00776166
\(383\) 2.80324 0.143239 0.0716194 0.997432i \(-0.477183\pi\)
0.0716194 + 0.997432i \(0.477183\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) 10.9237 0.556003
\(387\) −7.23852 −0.367955
\(388\) −14.5564 −0.738990
\(389\) 1.06755 0.0541267 0.0270634 0.999634i \(-0.491384\pi\)
0.0270634 + 0.999634i \(0.491384\pi\)
\(390\) 19.0477 0.964519
\(391\) 7.03079 0.355562
\(392\) 0 0
\(393\) −19.1405 −0.965511
\(394\) 13.6149 0.685909
\(395\) 46.9264 2.36112
\(396\) −1.26974 −0.0638066
\(397\) 5.00309 0.251098 0.125549 0.992087i \(-0.459931\pi\)
0.125549 + 0.992087i \(0.459931\pi\)
\(398\) −22.5698 −1.13132
\(399\) 0 0
\(400\) 7.09279 0.354640
\(401\) −22.7527 −1.13622 −0.568109 0.822953i \(-0.692325\pi\)
−0.568109 + 0.822953i \(0.692325\pi\)
\(402\) 4.37637 0.218274
\(403\) −52.4208 −2.61127
\(404\) 0.817512 0.0406727
\(405\) 3.47747 0.172797
\(406\) 0 0
\(407\) −2.17940 −0.108029
\(408\) 7.03079 0.348076
\(409\) −8.18558 −0.404751 −0.202375 0.979308i \(-0.564866\pi\)
−0.202375 + 0.979308i \(0.564866\pi\)
\(410\) 29.5872 1.46121
\(411\) −4.86215 −0.239832
\(412\) 4.50826 0.222106
\(413\) 0 0
\(414\) −1.00000 −0.0491473
\(415\) −17.7582 −0.871715
\(416\) −5.47747 −0.268555
\(417\) −16.5083 −0.808413
\(418\) 2.53947 0.124210
\(419\) −2.18558 −0.106773 −0.0533863 0.998574i \(-0.517001\pi\)
−0.0533863 + 0.998574i \(0.517001\pi\)
\(420\) 0 0
\(421\) −4.97475 −0.242455 −0.121227 0.992625i \(-0.538683\pi\)
−0.121227 + 0.992625i \(0.538683\pi\)
\(422\) 9.49441 0.462181
\(423\) −12.0169 −0.584284
\(424\) 7.68520 0.373226
\(425\) −49.8679 −2.41895
\(426\) 10.0616 0.487485
\(427\) 0 0
\(428\) −3.62363 −0.175155
\(429\) −6.95494 −0.335788
\(430\) −25.1717 −1.21389
\(431\) −16.1267 −0.776794 −0.388397 0.921492i \(-0.626971\pi\)
−0.388397 + 0.921492i \(0.626971\pi\)
\(432\) −1.00000 −0.0481125
\(433\) −15.5729 −0.748387 −0.374194 0.927351i \(-0.622080\pi\)
−0.374194 + 0.927351i \(0.622080\pi\)
\(434\) 0 0
\(435\) 25.7389 1.23409
\(436\) 18.3175 0.877247
\(437\) 2.00000 0.0956730
\(438\) 13.4944 0.644788
\(439\) 11.0308 0.526471 0.263235 0.964732i \(-0.415210\pi\)
0.263235 + 0.964732i \(0.415210\pi\)
\(440\) −4.41547 −0.210499
\(441\) 0 0
\(442\) 38.5109 1.83178
\(443\) −12.6600 −0.601493 −0.300746 0.953704i \(-0.597236\pi\)
−0.300746 + 0.953704i \(0.597236\pi\)
\(444\) −1.71642 −0.0814575
\(445\) −38.3592 −1.81840
\(446\) −15.2946 −0.724218
\(447\) −18.8257 −0.890426
\(448\) 0 0
\(449\) 3.07894 0.145304 0.0726521 0.997357i \(-0.476854\pi\)
0.0726521 + 0.997357i \(0.476854\pi\)
\(450\) 7.09279 0.334357
\(451\) −10.8032 −0.508705
\(452\) −10.5083 −0.494267
\(453\) −15.4632 −0.726524
\(454\) −1.42931 −0.0670810
\(455\) 0 0
\(456\) 2.00000 0.0936586
\(457\) −15.4659 −0.723462 −0.361731 0.932282i \(-0.617814\pi\)
−0.361731 + 0.932282i \(0.617814\pi\)
\(458\) −17.9241 −0.837541
\(459\) 7.03079 0.328169
\(460\) −3.47747 −0.162138
\(461\) −14.0196 −0.652958 −0.326479 0.945204i \(-0.605862\pi\)
−0.326479 + 0.945204i \(0.605862\pi\)
\(462\) 0 0
\(463\) 6.72153 0.312376 0.156188 0.987727i \(-0.450079\pi\)
0.156188 + 0.987727i \(0.450079\pi\)
\(464\) −7.40162 −0.343612
\(465\) −33.2803 −1.54334
\(466\) −18.4155 −0.853080
\(467\) 27.7104 1.28228 0.641141 0.767423i \(-0.278461\pi\)
0.641141 + 0.767423i \(0.278461\pi\)
\(468\) −5.47747 −0.253196
\(469\) 0 0
\(470\) −41.7885 −1.92756
\(471\) 12.8791 0.593437
\(472\) −7.84830 −0.361247
\(473\) 9.19101 0.422603
\(474\) −13.4944 −0.619819
\(475\) −14.1856 −0.650879
\(476\) 0 0
\(477\) 7.68520 0.351881
\(478\) −8.00000 −0.365911
\(479\) −16.0339 −0.732607 −0.366303 0.930495i \(-0.619377\pi\)
−0.366303 + 0.930495i \(0.619377\pi\)
\(480\) −3.47747 −0.158724
\(481\) −9.40162 −0.428677
\(482\) 4.39853 0.200347
\(483\) 0 0
\(484\) −9.38777 −0.426717
\(485\) −50.6195 −2.29851
\(486\) −1.00000 −0.0453609
\(487\) 24.8794 1.12739 0.563697 0.825982i \(-0.309379\pi\)
0.563697 + 0.825982i \(0.309379\pi\)
\(488\) −9.92415 −0.449245
\(489\) 23.1405 1.04645
\(490\) 0 0
\(491\) 34.9603 1.57773 0.788867 0.614563i \(-0.210668\pi\)
0.788867 + 0.614563i \(0.210668\pi\)
\(492\) −8.50826 −0.383582
\(493\) 52.0392 2.34373
\(494\) 10.9549 0.492886
\(495\) −4.41547 −0.198460
\(496\) 9.57026 0.429717
\(497\) 0 0
\(498\) 5.10664 0.228834
\(499\) 8.74166 0.391331 0.195665 0.980671i \(-0.437313\pi\)
0.195665 + 0.980671i \(0.437313\pi\)
\(500\) 7.27761 0.325465
\(501\) −17.4186 −0.778204
\(502\) −4.29498 −0.191694
\(503\) 6.76936 0.301831 0.150915 0.988547i \(-0.451778\pi\)
0.150915 + 0.988547i \(0.451778\pi\)
\(504\) 0 0
\(505\) 2.84287 0.126506
\(506\) 1.26974 0.0564466
\(507\) −17.0027 −0.755115
\(508\) 11.6149 0.515328
\(509\) 0.906783 0.0401925 0.0200962 0.999798i \(-0.493603\pi\)
0.0200962 + 0.999798i \(0.493603\pi\)
\(510\) 24.4493 1.08264
\(511\) 0 0
\(512\) 1.00000 0.0441942
\(513\) 2.00000 0.0883022
\(514\) −13.0451 −0.575393
\(515\) 15.6773 0.690825
\(516\) 7.23852 0.318658
\(517\) 15.2583 0.671061
\(518\) 0 0
\(519\) 2.84521 0.124891
\(520\) −19.0477 −0.835298
\(521\) 29.4296 1.28934 0.644668 0.764463i \(-0.276996\pi\)
0.644668 + 0.764463i \(0.276996\pi\)
\(522\) −7.40162 −0.323960
\(523\) 27.0789 1.18408 0.592040 0.805909i \(-0.298323\pi\)
0.592040 + 0.805909i \(0.298323\pi\)
\(524\) 19.1405 0.836157
\(525\) 0 0
\(526\) 13.4016 0.584338
\(527\) −67.2865 −2.93104
\(528\) 1.26974 0.0552582
\(529\) 1.00000 0.0434783
\(530\) 26.7251 1.16086
\(531\) −7.84830 −0.340587
\(532\) 0 0
\(533\) −46.6037 −2.01863
\(534\) 11.0308 0.477349
\(535\) −12.6010 −0.544791
\(536\) −4.37637 −0.189031
\(537\) −1.13785 −0.0491020
\(538\) 9.09236 0.392000
\(539\) 0 0
\(540\) −3.47747 −0.149646
\(541\) 10.4440 0.449023 0.224512 0.974471i \(-0.427921\pi\)
0.224512 + 0.974471i \(0.427921\pi\)
\(542\) −15.9241 −0.684001
\(543\) 18.9407 0.812822
\(544\) −7.03079 −0.301443
\(545\) 63.6984 2.72854
\(546\) 0 0
\(547\) 9.72429 0.415781 0.207890 0.978152i \(-0.433340\pi\)
0.207890 + 0.978152i \(0.433340\pi\)
\(548\) 4.86215 0.207701
\(549\) −9.92415 −0.423552
\(550\) −9.00597 −0.384016
\(551\) 14.8032 0.630639
\(552\) 1.00000 0.0425628
\(553\) 0 0
\(554\) −18.7527 −0.796728
\(555\) −5.96879 −0.253361
\(556\) 16.5083 0.700106
\(557\) −5.80921 −0.246144 −0.123072 0.992398i \(-0.539275\pi\)
−0.123072 + 0.992398i \(0.539275\pi\)
\(558\) 9.57026 0.405141
\(559\) 39.6488 1.67696
\(560\) 0 0
\(561\) −8.92724 −0.376908
\(562\) −24.4806 −1.03265
\(563\) −9.55247 −0.402588 −0.201294 0.979531i \(-0.564515\pi\)
−0.201294 + 0.979531i \(0.564515\pi\)
\(564\) 12.0169 0.506004
\(565\) −36.5421 −1.53734
\(566\) 6.18558 0.259999
\(567\) 0 0
\(568\) −10.0616 −0.422174
\(569\) −43.5476 −1.82561 −0.912804 0.408397i \(-0.866088\pi\)
−0.912804 + 0.408397i \(0.866088\pi\)
\(570\) 6.95494 0.291310
\(571\) 17.5335 0.733754 0.366877 0.930269i \(-0.380427\pi\)
0.366877 + 0.930269i \(0.380427\pi\)
\(572\) 6.95494 0.290801
\(573\) 0.151700 0.00633737
\(574\) 0 0
\(575\) −7.09279 −0.295790
\(576\) 1.00000 0.0416667
\(577\) 13.4944 0.561780 0.280890 0.959740i \(-0.409370\pi\)
0.280890 + 0.959740i \(0.409370\pi\)
\(578\) 32.4320 1.34899
\(579\) −10.9237 −0.453974
\(580\) −25.7389 −1.06875
\(581\) 0 0
\(582\) 14.5564 0.603383
\(583\) −9.75818 −0.404142
\(584\) −13.4944 −0.558403
\(585\) −19.0477 −0.787527
\(586\) 32.0196 1.32272
\(587\) −30.9264 −1.27647 −0.638234 0.769842i \(-0.720335\pi\)
−0.638234 + 0.769842i \(0.720335\pi\)
\(588\) 0 0
\(589\) −19.1405 −0.788671
\(590\) −27.2922 −1.12360
\(591\) −13.6149 −0.560042
\(592\) 1.71642 0.0705443
\(593\) −33.6765 −1.38293 −0.691463 0.722411i \(-0.743034\pi\)
−0.691463 + 0.722411i \(0.743034\pi\)
\(594\) 1.26974 0.0520979
\(595\) 0 0
\(596\) 18.8257 0.771132
\(597\) 22.5698 0.923722
\(598\) 5.47747 0.223990
\(599\) 25.0165 1.02215 0.511074 0.859537i \(-0.329248\pi\)
0.511074 + 0.859537i \(0.329248\pi\)
\(600\) −7.09279 −0.289562
\(601\) −9.73538 −0.397114 −0.198557 0.980089i \(-0.563626\pi\)
−0.198557 + 0.980089i \(0.563626\pi\)
\(602\) 0 0
\(603\) −4.37637 −0.178220
\(604\) 15.4632 0.629188
\(605\) −32.6457 −1.32724
\(606\) −0.817512 −0.0332091
\(607\) −3.69426 −0.149946 −0.0749728 0.997186i \(-0.523887\pi\)
−0.0749728 + 0.997186i \(0.523887\pi\)
\(608\) −2.00000 −0.0811107
\(609\) 0 0
\(610\) −34.5109 −1.39731
\(611\) 65.8224 2.66289
\(612\) −7.03079 −0.284203
\(613\) −7.65484 −0.309176 −0.154588 0.987979i \(-0.549405\pi\)
−0.154588 + 0.987979i \(0.549405\pi\)
\(614\) 20.1710 0.814034
\(615\) −29.5872 −1.19307
\(616\) 0 0
\(617\) 2.37116 0.0954594 0.0477297 0.998860i \(-0.484801\pi\)
0.0477297 + 0.998860i \(0.484801\pi\)
\(618\) −4.50826 −0.181349
\(619\) 27.8474 1.11928 0.559642 0.828735i \(-0.310939\pi\)
0.559642 + 0.828735i \(0.310939\pi\)
\(620\) 33.2803 1.33657
\(621\) 1.00000 0.0401286
\(622\) −8.10973 −0.325171
\(623\) 0 0
\(624\) 5.47747 0.219274
\(625\) −10.1563 −0.406251
\(626\) −6.84521 −0.273589
\(627\) −2.53947 −0.101417
\(628\) −12.8791 −0.513932
\(629\) −12.0678 −0.481173
\(630\) 0 0
\(631\) −24.8648 −0.989853 −0.494926 0.868935i \(-0.664805\pi\)
−0.494926 + 0.868935i \(0.664805\pi\)
\(632\) 13.4944 0.536779
\(633\) −9.49441 −0.377369
\(634\) −0.598381 −0.0237648
\(635\) 40.3904 1.60285
\(636\) −7.68520 −0.304738
\(637\) 0 0
\(638\) 9.39810 0.372074
\(639\) −10.0616 −0.398030
\(640\) 3.47747 0.137459
\(641\) 26.1267 1.03194 0.515971 0.856606i \(-0.327431\pi\)
0.515971 + 0.856606i \(0.327431\pi\)
\(642\) 3.62363 0.143013
\(643\) 18.8309 0.742620 0.371310 0.928509i \(-0.378909\pi\)
0.371310 + 0.928509i \(0.378909\pi\)
\(644\) 0 0
\(645\) 25.1717 0.991136
\(646\) 14.0616 0.553245
\(647\) 47.1539 1.85381 0.926906 0.375293i \(-0.122458\pi\)
0.926906 + 0.375293i \(0.122458\pi\)
\(648\) 1.00000 0.0392837
\(649\) 9.96527 0.391171
\(650\) −38.8505 −1.52384
\(651\) 0 0
\(652\) −23.1405 −0.906253
\(653\) 15.2610 0.597209 0.298605 0.954377i \(-0.403479\pi\)
0.298605 + 0.954377i \(0.403479\pi\)
\(654\) −18.3175 −0.716270
\(655\) 66.5606 2.60074
\(656\) 8.50826 0.332192
\(657\) −13.4944 −0.526467
\(658\) 0 0
\(659\) −24.9663 −0.972550 −0.486275 0.873806i \(-0.661645\pi\)
−0.486275 + 0.873806i \(0.661645\pi\)
\(660\) 4.41547 0.171872
\(661\) 7.09236 0.275861 0.137930 0.990442i \(-0.455955\pi\)
0.137930 + 0.990442i \(0.455955\pi\)
\(662\) 25.9099 1.00702
\(663\) −38.5109 −1.49564
\(664\) −5.10664 −0.198176
\(665\) 0 0
\(666\) 1.71642 0.0665098
\(667\) 7.40162 0.286592
\(668\) 17.4186 0.673944
\(669\) 15.2946 0.591322
\(670\) −15.2187 −0.587950
\(671\) 12.6010 0.486458
\(672\) 0 0
\(673\) −39.0358 −1.50472 −0.752360 0.658753i \(-0.771084\pi\)
−0.752360 + 0.658753i \(0.771084\pi\)
\(674\) −8.21328 −0.316364
\(675\) −7.09279 −0.273002
\(676\) 17.0027 0.653949
\(677\) 13.9580 0.536451 0.268225 0.963356i \(-0.413563\pi\)
0.268225 + 0.963356i \(0.413563\pi\)
\(678\) 10.5083 0.403567
\(679\) 0 0
\(680\) −24.4493 −0.937590
\(681\) 1.42931 0.0547714
\(682\) −12.1517 −0.465313
\(683\) −27.4667 −1.05098 −0.525492 0.850798i \(-0.676119\pi\)
−0.525492 + 0.850798i \(0.676119\pi\)
\(684\) −2.00000 −0.0764719
\(685\) 16.9080 0.646020
\(686\) 0 0
\(687\) 17.9241 0.683849
\(688\) −7.23852 −0.275966
\(689\) −42.0955 −1.60371
\(690\) 3.47747 0.132385
\(691\) 5.03046 0.191368 0.0956838 0.995412i \(-0.469496\pi\)
0.0956838 + 0.995412i \(0.469496\pi\)
\(692\) −2.84521 −0.108159
\(693\) 0 0
\(694\) −15.4632 −0.586975
\(695\) 57.4070 2.17757
\(696\) 7.40162 0.280558
\(697\) −59.8198 −2.26583
\(698\) 11.9241 0.451336
\(699\) 18.4155 0.696537
\(700\) 0 0
\(701\) 9.56120 0.361121 0.180561 0.983564i \(-0.442209\pi\)
0.180561 + 0.983564i \(0.442209\pi\)
\(702\) 5.47747 0.206734
\(703\) −3.43283 −0.129472
\(704\) −1.26974 −0.0478550
\(705\) 41.7885 1.57385
\(706\) −13.9688 −0.525722
\(707\) 0 0
\(708\) 7.84830 0.294957
\(709\) −31.5050 −1.18319 −0.591597 0.806234i \(-0.701502\pi\)
−0.591597 + 0.806234i \(0.701502\pi\)
\(710\) −34.9888 −1.31311
\(711\) 13.4944 0.506080
\(712\) −11.0308 −0.413396
\(713\) −9.57026 −0.358409
\(714\) 0 0
\(715\) 24.1856 0.904489
\(716\) 1.13785 0.0425235
\(717\) 8.00000 0.298765
\(718\) 6.56983 0.245184
\(719\) −3.38734 −0.126327 −0.0631633 0.998003i \(-0.520119\pi\)
−0.0631633 + 0.998003i \(0.520119\pi\)
\(720\) 3.47747 0.129598
\(721\) 0 0
\(722\) −15.0000 −0.558242
\(723\) −4.39853 −0.163583
\(724\) −18.9407 −0.703924
\(725\) −52.4981 −1.94973
\(726\) 9.38777 0.348413
\(727\) 25.0504 0.929068 0.464534 0.885555i \(-0.346222\pi\)
0.464534 + 0.885555i \(0.346222\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) −46.9264 −1.73682
\(731\) 50.8925 1.88233
\(732\) 9.92415 0.366807
\(733\) 15.3404 0.566610 0.283305 0.959030i \(-0.408569\pi\)
0.283305 + 0.959030i \(0.408569\pi\)
\(734\) 13.2499 0.489063
\(735\) 0 0
\(736\) −1.00000 −0.0368605
\(737\) 5.55684 0.204689
\(738\) 8.50826 0.313193
\(739\) −10.0616 −0.370121 −0.185061 0.982727i \(-0.559248\pi\)
−0.185061 + 0.982727i \(0.559248\pi\)
\(740\) 5.96879 0.219417
\(741\) −10.9549 −0.402440
\(742\) 0 0
\(743\) 22.5109 0.825846 0.412923 0.910766i \(-0.364508\pi\)
0.412923 + 0.910766i \(0.364508\pi\)
\(744\) −9.57026 −0.350863
\(745\) 65.4659 2.39848
\(746\) 19.6852 0.720726
\(747\) −5.10664 −0.186842
\(748\) 8.92724 0.326412
\(749\) 0 0
\(750\) −7.27761 −0.265741
\(751\) 30.4770 1.11212 0.556062 0.831141i \(-0.312312\pi\)
0.556062 + 0.831141i \(0.312312\pi\)
\(752\) −12.0169 −0.438213
\(753\) 4.29498 0.156518
\(754\) 40.5421 1.47646
\(755\) 53.7728 1.95699
\(756\) 0 0
\(757\) −25.3652 −0.921914 −0.460957 0.887423i \(-0.652494\pi\)
−0.460957 + 0.887423i \(0.652494\pi\)
\(758\) 9.77799 0.355153
\(759\) −1.26974 −0.0460885
\(760\) −6.95494 −0.252282
\(761\) 43.4043 1.57340 0.786702 0.617333i \(-0.211787\pi\)
0.786702 + 0.617333i \(0.211787\pi\)
\(762\) −11.6149 −0.420763
\(763\) 0 0
\(764\) −0.151700 −0.00548832
\(765\) −24.4493 −0.883968
\(766\) 2.80324 0.101285
\(767\) 42.9888 1.55224
\(768\) −1.00000 −0.0360844
\(769\) −9.32577 −0.336296 −0.168148 0.985762i \(-0.553779\pi\)
−0.168148 + 0.985762i \(0.553779\pi\)
\(770\) 0 0
\(771\) 13.0451 0.469806
\(772\) 10.9237 0.393153
\(773\) −12.6742 −0.455860 −0.227930 0.973677i \(-0.573196\pi\)
−0.227930 + 0.973677i \(0.573196\pi\)
\(774\) −7.23852 −0.260183
\(775\) 67.8798 2.43832
\(776\) −14.5564 −0.522545
\(777\) 0 0
\(778\) 1.06755 0.0382734
\(779\) −17.0165 −0.609680
\(780\) 19.0477 0.682018
\(781\) 12.7755 0.457145
\(782\) 7.03079 0.251420
\(783\) 7.40162 0.264512
\(784\) 0 0
\(785\) −44.7866 −1.59850
\(786\) −19.1405 −0.682720
\(787\) 16.7408 0.596745 0.298373 0.954449i \(-0.403556\pi\)
0.298373 + 0.954449i \(0.403556\pi\)
\(788\) 13.6149 0.485011
\(789\) −13.4016 −0.477110
\(790\) 46.9264 1.66957
\(791\) 0 0
\(792\) −1.26974 −0.0451181
\(793\) 54.3592 1.93035
\(794\) 5.00309 0.177553
\(795\) −26.7251 −0.947840
\(796\) −22.5698 −0.799967
\(797\) 29.6622 1.05069 0.525344 0.850890i \(-0.323936\pi\)
0.525344 + 0.850890i \(0.323936\pi\)
\(798\) 0 0
\(799\) 84.4886 2.98899
\(800\) 7.09279 0.250768
\(801\) −11.0308 −0.389754
\(802\) −22.7527 −0.803427
\(803\) 17.1343 0.604657
\(804\) 4.37637 0.154343
\(805\) 0 0
\(806\) −52.4208 −1.84644
\(807\) −9.09236 −0.320066
\(808\) 0.817512 0.0287600
\(809\) 13.2187 0.464745 0.232372 0.972627i \(-0.425351\pi\)
0.232372 + 0.972627i \(0.425351\pi\)
\(810\) 3.47747 0.122186
\(811\) −22.7554 −0.799051 −0.399525 0.916722i \(-0.630825\pi\)
−0.399525 + 0.916722i \(0.630825\pi\)
\(812\) 0 0
\(813\) 15.9241 0.558484
\(814\) −2.17940 −0.0763878
\(815\) −80.4704 −2.81876
\(816\) 7.03079 0.246127
\(817\) 14.4770 0.506488
\(818\) −8.18558 −0.286202
\(819\) 0 0
\(820\) 29.5872 1.03323
\(821\) −21.6184 −0.754488 −0.377244 0.926114i \(-0.623128\pi\)
−0.377244 + 0.926114i \(0.623128\pi\)
\(822\) −4.86215 −0.169587
\(823\) −19.4008 −0.676268 −0.338134 0.941098i \(-0.609796\pi\)
−0.338134 + 0.941098i \(0.609796\pi\)
\(824\) 4.50826 0.157053
\(825\) 9.00597 0.313548
\(826\) 0 0
\(827\) −37.5388 −1.30535 −0.652677 0.757637i \(-0.726354\pi\)
−0.652677 + 0.757637i \(0.726354\pi\)
\(828\) −1.00000 −0.0347524
\(829\) 33.6823 1.16984 0.584918 0.811093i \(-0.301127\pi\)
0.584918 + 0.811093i \(0.301127\pi\)
\(830\) −17.7582 −0.616395
\(831\) 18.7527 0.650526
\(832\) −5.47747 −0.189897
\(833\) 0 0
\(834\) −16.5083 −0.571634
\(835\) 60.5725 2.09620
\(836\) 2.53947 0.0878295
\(837\) −9.57026 −0.330797
\(838\) −2.18558 −0.0754996
\(839\) −34.0330 −1.17495 −0.587475 0.809242i \(-0.699878\pi\)
−0.587475 + 0.809242i \(0.699878\pi\)
\(840\) 0 0
\(841\) 25.7840 0.889102
\(842\) −4.97475 −0.171441
\(843\) 24.4806 0.843155
\(844\) 9.49441 0.326811
\(845\) 59.1262 2.03400
\(846\) −12.0169 −0.413151
\(847\) 0 0
\(848\) 7.68520 0.263911
\(849\) −6.18558 −0.212289
\(850\) −49.8679 −1.71046
\(851\) −1.71642 −0.0588380
\(852\) 10.0616 0.344704
\(853\) −29.6630 −1.01564 −0.507822 0.861462i \(-0.669549\pi\)
−0.507822 + 0.861462i \(0.669549\pi\)
\(854\) 0 0
\(855\) −6.95494 −0.237854
\(856\) −3.62363 −0.123853
\(857\) −37.0200 −1.26458 −0.632290 0.774732i \(-0.717885\pi\)
−0.632290 + 0.774732i \(0.717885\pi\)
\(858\) −6.95494 −0.237438
\(859\) −1.24992 −0.0426467 −0.0213233 0.999773i \(-0.506788\pi\)
−0.0213233 + 0.999773i \(0.506788\pi\)
\(860\) −25.1717 −0.858349
\(861\) 0 0
\(862\) −16.1267 −0.549277
\(863\) −56.5953 −1.92653 −0.963263 0.268559i \(-0.913452\pi\)
−0.963263 + 0.268559i \(0.913452\pi\)
\(864\) −1.00000 −0.0340207
\(865\) −9.89412 −0.336410
\(866\) −15.5729 −0.529190
\(867\) −32.4320 −1.10145
\(868\) 0 0
\(869\) −17.1343 −0.581242
\(870\) 25.7389 0.872630
\(871\) 23.9715 0.812242
\(872\) 18.3175 0.620308
\(873\) −14.5564 −0.492660
\(874\) 2.00000 0.0676510
\(875\) 0 0
\(876\) 13.4944 0.455934
\(877\) −35.7139 −1.20597 −0.602986 0.797752i \(-0.706022\pi\)
−0.602986 + 0.797752i \(0.706022\pi\)
\(878\) 11.0308 0.372271
\(879\) −32.0196 −1.07999
\(880\) −4.41547 −0.148845
\(881\) 37.5306 1.26444 0.632219 0.774789i \(-0.282144\pi\)
0.632219 + 0.774789i \(0.282144\pi\)
\(882\) 0 0
\(883\) −32.7242 −1.10126 −0.550629 0.834750i \(-0.685612\pi\)
−0.550629 + 0.834750i \(0.685612\pi\)
\(884\) 38.5109 1.29526
\(885\) 27.2922 0.917418
\(886\) −12.6600 −0.425320
\(887\) −4.18792 −0.140616 −0.0703082 0.997525i \(-0.522398\pi\)
−0.0703082 + 0.997525i \(0.522398\pi\)
\(888\) −1.71642 −0.0575992
\(889\) 0 0
\(890\) −38.3592 −1.28580
\(891\) −1.26974 −0.0425377
\(892\) −15.2946 −0.512100
\(893\) 24.0339 0.804263
\(894\) −18.8257 −0.629626
\(895\) 3.95685 0.132263
\(896\) 0 0
\(897\) −5.47747 −0.182887
\(898\) 3.07894 0.102746
\(899\) −70.8354 −2.36249
\(900\) 7.09279 0.236426
\(901\) −54.0330 −1.80010
\(902\) −10.8032 −0.359708
\(903\) 0 0
\(904\) −10.5083 −0.349499
\(905\) −65.8656 −2.18945
\(906\) −15.4632 −0.513730
\(907\) 39.6097 1.31522 0.657609 0.753359i \(-0.271568\pi\)
0.657609 + 0.753359i \(0.271568\pi\)
\(908\) −1.42931 −0.0474335
\(909\) 0.817512 0.0271152
\(910\) 0 0
\(911\) −22.3843 −0.741623 −0.370812 0.928708i \(-0.620920\pi\)
−0.370812 + 0.928708i \(0.620920\pi\)
\(912\) 2.00000 0.0662266
\(913\) 6.48408 0.214592
\(914\) −15.4659 −0.511565
\(915\) 34.5109 1.14090
\(916\) −17.9241 −0.592231
\(917\) 0 0
\(918\) 7.03079 0.232051
\(919\) −32.6010 −1.07541 −0.537705 0.843133i \(-0.680708\pi\)
−0.537705 + 0.843133i \(0.680708\pi\)
\(920\) −3.47747 −0.114649
\(921\) −20.1710 −0.664656
\(922\) −14.0196 −0.461711
\(923\) 55.1120 1.81403
\(924\) 0 0
\(925\) 12.1742 0.400285
\(926\) 6.72153 0.220883
\(927\) 4.50826 0.148071
\(928\) −7.40162 −0.242970
\(929\) −58.4235 −1.91681 −0.958406 0.285409i \(-0.907871\pi\)
−0.958406 + 0.285409i \(0.907871\pi\)
\(930\) −33.2803 −1.09130
\(931\) 0 0
\(932\) −18.4155 −0.603219
\(933\) 8.10973 0.265501
\(934\) 27.7104 0.906710
\(935\) 31.0442 1.01525
\(936\) −5.47747 −0.179037
\(937\) −20.3431 −0.664581 −0.332291 0.943177i \(-0.607821\pi\)
−0.332291 + 0.943177i \(0.607821\pi\)
\(938\) 0 0
\(939\) 6.84521 0.223385
\(940\) −41.7885 −1.36299
\(941\) 39.4221 1.28512 0.642562 0.766234i \(-0.277872\pi\)
0.642562 + 0.766234i \(0.277872\pi\)
\(942\) 12.8791 0.419623
\(943\) −8.50826 −0.277067
\(944\) −7.84830 −0.255440
\(945\) 0 0
\(946\) 9.19101 0.298825
\(947\) −45.5587 −1.48046 −0.740229 0.672355i \(-0.765283\pi\)
−0.740229 + 0.672355i \(0.765283\pi\)
\(948\) −13.4944 −0.438278
\(949\) 73.9152 2.39939
\(950\) −14.1856 −0.460241
\(951\) 0.598381 0.0194038
\(952\) 0 0
\(953\) 14.0782 0.456037 0.228019 0.973657i \(-0.426775\pi\)
0.228019 + 0.973657i \(0.426775\pi\)
\(954\) 7.68520 0.248818
\(955\) −0.527533 −0.0170706
\(956\) −8.00000 −0.258738
\(957\) −9.39810 −0.303797
\(958\) −16.0339 −0.518031
\(959\) 0 0
\(960\) −3.47747 −0.112235
\(961\) 60.5899 1.95451
\(962\) −9.40162 −0.303120
\(963\) −3.62363 −0.116770
\(964\) 4.39853 0.141667
\(965\) 37.9869 1.22284
\(966\) 0 0
\(967\) 45.5102 1.46351 0.731754 0.681569i \(-0.238702\pi\)
0.731754 + 0.681569i \(0.238702\pi\)
\(968\) −9.38777 −0.301734
\(969\) −14.0616 −0.451723
\(970\) −50.6195 −1.62529
\(971\) −30.0053 −0.962917 −0.481458 0.876469i \(-0.659893\pi\)
−0.481458 + 0.876469i \(0.659893\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 0 0
\(974\) 24.8794 0.797188
\(975\) 38.8505 1.24421
\(976\) −9.92415 −0.317664
\(977\) 49.0531 1.56935 0.784673 0.619910i \(-0.212831\pi\)
0.784673 + 0.619910i \(0.212831\pi\)
\(978\) 23.1405 0.739952
\(979\) 14.0062 0.447640
\(980\) 0 0
\(981\) 18.3175 0.584832
\(982\) 34.9603 1.11563
\(983\) −25.7582 −0.821558 −0.410779 0.911735i \(-0.634743\pi\)
−0.410779 + 0.911735i \(0.634743\pi\)
\(984\) −8.50826 −0.271233
\(985\) 47.3454 1.50855
\(986\) 52.0392 1.65727
\(987\) 0 0
\(988\) 10.9549 0.348523
\(989\) 7.23852 0.230172
\(990\) −4.41547 −0.140333
\(991\) −15.9715 −0.507350 −0.253675 0.967290i \(-0.581639\pi\)
−0.253675 + 0.967290i \(0.581639\pi\)
\(992\) 9.57026 0.303856
\(993\) −25.9099 −0.822225
\(994\) 0 0
\(995\) −78.4859 −2.48817
\(996\) 5.10664 0.161810
\(997\) 15.8956 0.503419 0.251709 0.967803i \(-0.419007\pi\)
0.251709 + 0.967803i \(0.419007\pi\)
\(998\) 8.74166 0.276712
\(999\) −1.71642 −0.0543050
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 6762.2.a.cj.1.4 4
7.6 odd 2 6762.2.a.cs.1.1 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6762.2.a.cj.1.4 4 1.1 even 1 trivial
6762.2.a.cs.1.1 yes 4 7.6 odd 2